3.104 \(\int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx\)

Optimal. Leaf size=545 \[ \frac{d x \sqrt{a+b x^2} \sqrt{e+f x^2}}{2 f \sqrt{c+d x^2}}+\frac{b \sqrt{e} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right )|\frac{(b c-a d) e}{c (b e-a f)}\right )}{2 d f \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac{\sqrt{e} \sqrt{a+b x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{2 f \sqrt{e+f x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} (-a d f-b c f+b d e) \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{2 a d f \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

[Out]

(d*x*Sqrt[a + b*x^2]*Sqrt[e + f*x^2])/(2*f*Sqrt[c + d*x^2]) - (Sqrt[e]*Sqrt[d*e
- c*f]*Sqrt[a + b*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]*EllipticE[ArcSin[(S
qrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))
])/(2*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*x^2]) + (b*Sqrt[e]*(d*e
- c*f)*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticF[ArcSin[(
Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])], ((b*c - a*d)*e)/(c*(b*e - a*f))])
/(2*d*f*Sqrt[b*e - a*f]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2]) -
(c*Sqrt[e]*(b*d*e - b*c*f - a*d*f)*Sqrt[a + b*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c +
d*x^2))]*EllipticPi[(d*e)/(d*e - c*f), ArcSin[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt
[c + d*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(2*a*d*f*Sqrt[d*e - c*f]*Sqr
t[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 1.64071, antiderivative size = 545, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.206 \[ \frac{d x \sqrt{a+b x^2} \sqrt{e+f x^2}}{2 f \sqrt{c+d x^2}}+\frac{b \sqrt{e} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right )|\frac{(b c-a d) e}{c (b e-a f)}\right )}{2 d f \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac{\sqrt{e} \sqrt{a+b x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{2 f \sqrt{e+f x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} (-a d f-b c f+b d e) \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{2 a d f \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/Sqrt[e + f*x^2],x]

[Out]

(d*x*Sqrt[a + b*x^2]*Sqrt[e + f*x^2])/(2*f*Sqrt[c + d*x^2]) - (Sqrt[e]*Sqrt[d*e
- c*f]*Sqrt[a + b*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]*EllipticE[ArcSin[(S
qrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))
])/(2*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*x^2]) + (b*Sqrt[e]*(d*e
- c*f)*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticF[ArcSin[(
Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])], ((b*c - a*d)*e)/(c*(b*e - a*f))])
/(2*d*f*Sqrt[b*e - a*f]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2]) -
(c*Sqrt[e]*(b*d*e - b*c*f - a*d*f)*Sqrt[a + b*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c +
d*x^2))]*EllipticPi[(d*e)/(d*e - c*f), ArcSin[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt
[c + d*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(2*a*d*f*Sqrt[d*e - c*f]*Sqr
t[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*x^2])

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.120747, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{\sqrt{e+f x^2}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/Sqrt[e + f*x^2],x]

[Out]

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/Sqrt[e + f*x^2], x]

_______________________________________________________________________________________

Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{1\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{f{x}^{2}+e}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)

[Out]

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{\sqrt{f x^{2} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/sqrt(f*x^2 + e),x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/sqrt(f*x^2 + e), x)

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/sqrt(f*x^2 + e),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{\sqrt{e + f x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)

[Out]

Integral(sqrt(a + b*x**2)*sqrt(c + d*x**2)/sqrt(e + f*x**2), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{\sqrt{f x^{2} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/sqrt(f*x^2 + e),x, algorithm="giac")

[Out]

integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/sqrt(f*x^2 + e), x)